The polynomial partition of unity method: an efficient tool in approximation

Document Type : Research Paper

Authors

1 Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, Isfahan University, Isfahan, Iran

2 Department of Computer Science, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

3 Department of Mathematics, Khansar Campus, University of Isfahan, Iran

Abstract

In this paper, we investigate a polynomial approximation/interpolation based on the partition of unity method, and employ it as an efficient method for numerical solution of multivariate problems. First, we introduce multivariate polynomial approximations and prove their scalability properties. Then we use these properties to derive stability and convergence bounds for the proposed method. To have a stable algorithm, local approximations are computed over small subdomains and joined via the partition of unity weight functions to obtain a smooth global approximation. Finally, the global error bound is derived in terms of the error bounds of the local approximations. The idea behind this approach is to solve several small well-conditioned problems instead of solving one large ill-conditioned problem. From the computational point of view, this approach is highly efficient and applicable to a wide range of problems. As an example, the numerical solutions of differential equations are obtained using this approach. Instead of employing a background mesh (similar to finite element and finite volume methods), the unknown quantities are expressed in terms of scattered points. Hence, the given method can also be considered a so-called meshless method

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References

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Mathematics and Society
Volume 5, Issue 3 - Serial Number 3
September 2020
Pages 75-98

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History

  • Receive Date: 02 March 2021
  • Revise Date: 20 July 2021
  • Accept Date: 23 July 2021
  • Publish Date: 21 November 2020

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